%I #23 Aug 10 2020 02:01:21

%S 1,1,3,10,37,144,589,2483,10746,47420,212668,966324,4439540,20587286,

%T 96237484,453012296,2145478716,10215922013,48877938369,234862013473,

%U 1132902329028,5483947191651,26630419098206,129696204701807,633339363924611,3100369991303297

%N Number of objects generated by the Combstruct grammar defined in the Maple program. See the link for the grammar specification.

%C Number of free pure symmetric multifunctions with n + 1 unlabeled leaves. A free pure symmetric multifunction f in PSM is either (case 1) f = the leaf symbol "o", or (case 2) f = an expression of the form h[g_1, ..., g_k] where k > 0, h is in PSM, each of the g_i for i = 1, ..., k is in PSM, and for i < j we have g_i <= g_j under a canonical total ordering of PSM, such as the Mathematica ordering of expressions. - _Gus Wiseman_, Aug 02 2018

%H Alois P. Heinz, <a href="/A052893/b052893.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=869">Encyclopedia of Combinatorial Structures 869</a>

%H Maplesoft, <a href="https://www.maplesoft.com/support/help/Maple/view.aspx?path=examples%2fcombstruct_grammars">Combstruct grammars</a>.

%H Mathematica Reference, <a href="http://reference.wolfram.com/mathematica/ref/Orderless.html">Orderless</a>.

%F G.f.: 1/(1 - g(x)) where g(x) is the g.f. of A052891. - _Andrew Howroyd_, Aug 09 2020

%e From _Gus Wiseman_, Aug 02 2018: (Start)

%e The a(3) = 10 free pure symmetric multifunctions with 4 unlabeled leaves:

%e o[o[o[o]]]

%e o[o[o][o]]

%e o[o][o[o]]

%e o[o[o]][o]

%e o[o][o][o]

%e o[o[o,o]]

%e o[o,o[o]]

%e o[o][o,o]

%e o[o,o][o]

%e o[o,o,o]

%e (End)

%p spec := [S, {C = Set(B,1 <= card), B=Prod(Z,S), S=Sequence(C)}, unlabeled]:

%p seq(combstruct[count](spec, size=n), n=0..20);

%t multing[t_,n_]:=Array[(t+#-1)/#&,n,1,Times];

%t a[n_]:=a[n]=If[n==1,1,Sum[a[k]*Sum[Product[multing[a[First[s]],Length[s]],{s,Split[p]}],{p,IntegerPartitions[n-k]}],{k,1,n-1}]];

%t Array[a,30] (* _Gus Wiseman_, Aug 02 2018 *)

%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o seq(n)={my(v=[1]); for(n=1, n, v=Vec(1/(1-x*Ser(EulerT(v))))); v} \\ _Andrew Howroyd_, Aug 09 2020

%Y Cf. A001003, A052891, A277996, A279944, A280000.

%K easy,nonn

%O 0,3

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%E More terms from _Gus Wiseman_, Aug 02 2018